0

Given spectra $X$ and $Y$, the set $[X,Y]$ of homotopy classes of maps from $X$ to $Y$ can be endowed with an abelian group structure. Can the group $[X,Y]$ be expressed in terms of the homotopy groups $\pi_i(X)$ and $\pi_i(Y)$? A naive guess is\begin{equation*}[X,Y]\stackrel{?}{=}\bigoplus^\infty_{i=0}\text{Hom}(\pi_i(X),\pi_i(Y))\end{equation*}but this is not quite right. What is the correct expression, as general as possible?

Community
  • 1
Alex Turzillo
  • 449
  • "the set $[X,Y]$ of homotopy classes of maps from $X$ to $Y$ can be endowed with an abelian group structure." How do you propose to do this? I certainly do not see a natural way to do this (even if we constrict to pointed spaces) . What abelian group is $[S^1,S^1\vee S^1]$? What group is $[S^1 \vee S^1,S^1 \vee S^1]$? – PVAL-inactive Aug 18 '14 at 02:03
  • 1
    This can't work for spaces since $[X,Y]$ is in general only a set. If you work with spectra instead of spaces (the very open) Freyd's generating hypothesis is that for finite spectra $X$ and $Y$ the natural map $[X,Y] \to \text{Hom}{\pi* S}(\pi_X,\pi_ Y)$ is a monomorphism (which Freyd shows actually implies it is an isomorphism). – Drew Aug 18 '14 at 02:05
  • How do you define the composite of two elements in $[X,Y]$? – Hamou Aug 18 '14 at 02:05
  • @Drew is right. I am actually interested in spectra, in which case a group structure does exist. I'll edit the question. – Alex Turzillo Aug 18 '14 at 02:08
  • The generating hypothesis is exactly what I was looking for. Thanks! – Alex Turzillo Aug 18 '14 at 02:13
  • The naive guess is much too naive; if it were true then there would be essentially no difference between spectra and sequences (not even chain complexes) of abelian groups. In particular the naive guess would imply that there are no interesting maps between Eilenberg-MacLane spectra, so in particular no interesting stable cohomology operations. The generating hypothesis fixes things by restricting to finite spectra and keeping track of a natural source of extra structure on the homotopy groups. – Qiaochu Yuan Aug 18 '14 at 05:27

1 Answers1

1

Since you are interested in spectra then the generating hypothesis is what you want. Namely Freyd conjectured that for finite spectra $X$ and $Y$ the natural map $$ [X,Y] \to \text{Hom}_{\pi_* S}(\pi_*X,\pi_*Y) $$ is a monomorphism. I recommend Hovey's article for some equivalent statements and consequences.

Drew
  • 918